First, length-extension attacks apply to hash functions, not encryption algorithms. You don't need to worry about them here.
To answer your question, standard encryption algorithms (such as CBC) are secure against chosen-plaintext attacks, so in fact they are explicitly designed to be secure in this type of scenario---the ability to encrypt plaintexts of his choosing under an unknown key does not give an attacker useful information about that key (or about other ciphertexts).
If there is any possibility that an attacker can tamper with the ciphertexts (for example, you're sending them through an unauthenticated channel or storing them in untrusted storage), then you should follow the advice of CodesinChaos and also look into using authenticated encryption. Look up GCM or HMAC.
You said you'd be willing to accept this answer if I moved it from a comment, but I'd encourage you to take a close look at AJ Henderson's answer, too. The unusual requirements of this vaguely defined theoretical service leave me feeling a bit wary of making specific recommendations, but if his suggestion is practical, it's probably the way to go.